STEP 1: find angle $ \beta $
To find angle $ \beta $ use Thw Law of Sines:
$$ \dfrac{ \sin( \beta )} { b } = \dfrac{ \sin( \gamma )} { c } $$After substituting $b = 50$, $c = 100$ and $\gamma = 28^o$ we have:
$$ \dfrac{ \sin( \beta )} { 50 } = \dfrac{ \sin( 28^o )} { 100 } $$ $$ \dfrac{ \sin( \beta )} { 50 } = \dfrac{ 0.4695 } { 100 } $$ $$ \sin( \beta ) \cdot 100 = 50 \cdot 0.4695 $$ $$ \sin( \beta ) \cdot 100 = 23.4736 $$ $$ \sin( \beta ) = \dfrac{ 23.4736 }{ 100 } $$ $$ \sin( \beta ) = 0.2347 $$ $$ \beta = \arcsin{ \left( 0.2347 \right)} $$ $$ \beta \approx 13.576^o $$STEP 2: find angle $ \alpha $
To find angle $ \alpha $ use formula:
$$ \alpha + \beta + \gamma = 180^o $$After substituting $ \beta = 13.576^o $ and $ \gamma = 28^o $ we have:
$$ \alpha + 13.576^o + 28^o = 180^o $$ $$ \alpha + 41.576^o = 180^o $$ $$ \alpha = 180^o - 41.576^o $$ $$ \alpha = 138.424^o $$STEP 3: find side $ a $
To find side $ a $ use Law of Cosines:
$$ a^2 = b^2 + c^2 - 2 \cdot b \cdot c \cdot \cos( \alpha ) $$After substituting $b = 50$, $c = 100$ and $\alpha = 138.424^o$ we have:
$$ a^2 = 50^2 + 100^2 - 2 \cdot 50 \cdot 100 \cdot \cos( 138.424^o ) $$ $$ a^2 = 2500 + 10000 - 2 \cdot 50 \cdot 100 \cdot \cos( 138.424^o ) $$ $$ a^2 = 12500 - 2 \cdot 5000 \cdot \cos( 138.424^o ) $$ $$ a^2 = 12500 - 10000 \cdot \left(-0.7481\right) $$ $$ a^2 = 12500 - \left(-7480.7557\right) $$ $$ a^2 = 19980.7557 $$ $$ a = \sqrt{ 19980.7557 } $$$$ a \approx 141.3533 $$